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SUMMARY:Algebraic correspondences: Where rational dynamics meets Kleinian 
 groups
DTSTART:20261009T083000Z
DTEND:20261009T103000Z
DTSTAMP:20261021T194300Z
UID:indico-event-17381@indico.math.cnrs.fr
DESCRIPTION:Speakers: Luna Lomonaco\n\nThe analogies between the iteration
  of holomorphic maps and the actionof Kleinian groups were first systemati
 cally explored by DennisSullivan in the mid-1980s. In the landmark paper\,
  where he famouslyproved Fatou's conjecture—that rational maps on the Ri
 emann spherehave no wandering domains—Sullivan introduced what is now kn
 own asSullivan's Dictionary. This conceptual framework draws deep parallel
 sbetween the definitions\, theorems\, and conjectures of holomorphicdynami
 cs and those of Kleinian group theory.\n \nSullivan emphasized striking s
 imilarities between the Fatou set $F_f$and Julia set $J_f$ of a holomorphi
 c map $f$ on the Riemann sphere$\\widehat{\\mathbb{C}}$\, and the ordinary
  set $\\Omega(G)$ and limit set$\\Lambda(G)$ of a finitely generated Klein
 ian group $G$ acting on$\\widehat{\\mathbb{C}}$. His proof of the no wande
 ring domains theoremwas directly inspired by methods used to establish Ahl
 fors' FinitenessTheorem in the setting of Kleinian groups\, highlighting t
 he profoundconceptual bridges between the two fields.\n \nBoth rational m
 aps and finitely generated Kleinian groups can beregarded as special cases
  of holomorphic correspondences. An$n$-to-$m$ holomorphic correspondence o
 n $\\widehat{\\mathbb{C}}$ is amultivalued map $\\mathcal{F}: z \\mapsto w
 $ defined implicitly by apolynomial relation $P(z\, w) = 0$.\n \nIn 1994\
 , Shaun Bullett and Christopher Penrose introduced the firstfamily of corr
 espondences that contains matings between quadraticrational maps and the m
 odular group\, and proved that\, for a particularparameter\, the correspon
 dence is a mating : it behaves as the modulargroup on an open subset \\Ome
 ga\, and as a polynomial (and its inverse)in the complement.\nSince then\,
  the field of correspondences which are matings betweenrational maps and K
 leinian groups grew considerably.\n \nIn this talk\, I will give an overv
 iew of the subject.\n\nhttps://indico.math.cnrs.fr/event/17381/
URL:https://indico.math.cnrs.fr/event/17381/
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